Question:
Mark the correct alternative in the following:
The function $f(x)=\frac{\lambda \sin x+2 \cos x}{\sin x+\cos x}$ is increasing, if
A. $\lambda<1$
B. $\lambda>1$
C. $\lambda<2$
D. $\lambda>2$
Solution:
Formula:- (i) The necessary and sufficient condition for differentiable function defined on (a,b) to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)>0$ for all $x \in(a, b)$
Given:-
$f(x)=\frac{\lambda \sin x+2 \cos x}{\sin x+\cos x}$
For increasing function $f^{\prime}(x)<0$
$\frac{\mathrm{d}(\mathrm{f}(\mathrm{x}))}{\mathrm{dx}}=\mathrm{f}^{\prime}(\mathrm{x})=\frac{\lambda-2}{(\sin \mathrm{x}+\cos \mathrm{x})^{2}}>0$
$\Rightarrow \lambda>2$